Theorem eubid

Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994) (Proof shortened by Wolf Lammen, 19-Feb-2023)

	Ref	Expression
Hypotheses	eubid.1	|- F/ x ph
	eubid.2	|- ( ph -> ( ps <-> ch ) )
Assertion	eubid	|- ( ph -> ( E! x ps <-> E! x ch ) )

Proof

Step	Hyp	Ref	Expression
1		eubid.1	|- F/ x ph
2		eubid.2	|- ( ph -> ( ps <-> ch ) )
3	1 2	alrimi	|- ( ph -> A. x ( ps <-> ch ) )
4		eubi	|- ( A. x ( ps <-> ch ) -> ( E! x ps <-> E! x ch ) )
5	3 4	syl	|- ( ph -> ( E! x ps <-> E! x ch ) )